The one-dimensional propagation of a rarefaction wave is mathematically described by a system of two partial differential equations involving the velocity \(u\) of the gas and the heat flux \(q\). It is customary to study two approximations of the original system. The first is obtained by neglecting the partial derivatives \(q_x\), \(q_{xx}\) with respect to the \(x\)-direction of propagation while the second is obtained by dropping only \(q_{xx}\). The authors study this second approximation.
The first step in the treatment of the problem is the introduction of a potential function \(w(x, t)\) such that the original system can be reduced to a single partial differential equation in \(w\), with a step function as initial condition. The authors derive a set of pointwise estimates for \(w(x, t)\), provided that it is smooth. Otherwise they derive estimates in \(L^p\), where \(L^p\) is the usual Lebesgue space. Other estimates can be obtained for perturbed solutions of the smooth \(w(x, t)\) defined above.
Their main result is that, if the jump of the initial \(u\) is sufficiently small, then the initial value problem has a unique solution with a precise asymptotic behaviour for large times.